1. CSE373: Data Structures and Algorithms Lecture 2: Proof by Induction
Catie Baker
Spring 2015

2. Background on Induction
Type of mathematical proof
Typically used to establish a given statement for all natural numbers (e.g. integers > 0)
Proof is a sequence of deductive steps
Show the statement is true for the first number.
Show that if the statement is true for any one number, this implies the statement is true for the next number.
If so, we can infer that the statement is true for all numbers.
Spring 2015
CSE 373: Data Structures and Algorithms

3. Think about climbing a ladder
1. Show you can get to the first rung (base case)
2. Show you can get between rungs (inductive step)
3. Now you can climb forever.
Spring 2015
CSE 373: Data Structures and Algorithms

4. CSE 373: Data Structures and Algorithms
Why you should care
Induction turns out to be a useful technique
AVL trees
Heaps
Graph algorithms
Can also prove things like 3n > n3 for n ≥ 4
Exposure to rigorous thinking
Spring 2015
CSE 373: Data Structures and Algorithms

5. CSE 373: Data Structures and Algorithms
Example problem
Find the sum of the integers from 1 to n
… + (n-1) + n
For any n ≥ 1
Could use brute force, but would be slow
There’s probably a clever shortcut
Spring 2015
CSE 373: Data Structures and Algorithms

6. CSE 373: Data Structures and Algorithms
Finding the formula
Shortcut will be some formula involving n
Compare examples and look for patterns
Not something I will ask you to do!
Start with n = 10:
Large enough to be a pain to add up
Worthwhile to find shortcut
Spring 2015
CSE 373: Data Structures and Algorithms

7. CSE 373: Data Structures and Algorithms
Finding the formula
= 5×11
Spring 2015
CSE 373: Data Structures and Algorithms

8. CSE 373: Data Structures and Algorithms
Finding the formula
= 4×10 + 5
Spring 2015
CSE 373: Data Structures and Algorithms

9. CSE 373: Data Structures and Algorithms
Finding the formula
= 4×9
Spring 2015
CSE 373: Data Structures and Algorithms

10. CSE 373: Data Structures and Algorithms
Finding the formula
= 3×8 + 4
Spring 2015
CSE 373: Data Structures and Algorithms

11. CSE 373: Data Structures and Algorithms
Finding the formula
n=7
3×8 + 4
n=8
4×9
n=9
4×10 + 5
n=10
5×11
Spring 2015
CSE 373: Data Structures and Algorithms

12. CSE 373: Data Structures and Algorithms
Finding the formula
n=7
3×8 + 4
n is odd
n=8
4×9
n is even
n=9
4×10 + 5
n=10
5×11
Spring 2015
CSE 373: Data Structures and Algorithms

13. CSE 373: Data Structures and Algorithms
Finding the formula
When n is even
… + (n-3) + (n-2) + (n-1) + n
= (n/2)×(n+1)
Spring 2015
CSE 373: Data Structures and Algorithms

14. CSE 373: Data Structures and Algorithms
Finding the formula
3×8 + 4
4×9
n(n+1)/2
4×10 + 5
5×11
Spring 2015
CSE 373: Data Structures and Algorithms

15. CSE 373: Data Structures and Algorithms
Finding the formula
When n is odd
… + (n-3) + (n-2) + (n-1) + n
= ((n-1)/2)×(n+1) + (n+1)/2
Spring 2015
CSE 373: Data Structures and Algorithms

16. CSE 373: Data Structures and Algorithms
Finding the formula
When n is odd
… + (n-3) + (n-2) + (n-1) + n
= ((n-1)/2)×(n+1) + (n+1)/2
Spring 2015
CSE 373: Data Structures and Algorithms

17. CSE 373: Data Structures and Algorithms
Finding the formula
When n is odd
… + (n-3) + (n-2) + (n-1) + n
= ((n-1)×(n+1) + (n+1))/2
Spring 2015
CSE 373: Data Structures and Algorithms

18. CSE 373: Data Structures and Algorithms
Finding the formula
When n is odd
… + (n-3) + (n-2) + (n-1) + n
= ((n-1)×(n+1) + (n+1))/2
Spring 2015
CSE 373: Data Structures and Algorithms

19. CSE 373: Data Structures and Algorithms
Finding the formula
When n is odd
… + (n-3) + (n-2) + (n-1) + n
= ((n-1 + 1)×(n+1))/2
Spring 2015
CSE 373: Data Structures and Algorithms

20. CSE 373: Data Structures and Algorithms
Finding the formula
When n is odd
… + (n-3) + (n-2) + (n-1) + n
= ((n-1 + 1)×(n+1))/2
Spring 2015
CSE 373: Data Structures and Algorithms

21. CSE 373: Data Structures and Algorithms
Finding the formula
When n is odd
… + (n-3) + (n-2) + (n-1) + n
= (n (n+1))/2
Spring 2015
CSE 373: Data Structures and Algorithms

22. CSE 373: Data Structures and Algorithms
Finding the formula
3×8 + 4
n(n+1)/2
4×9
4×10 + 5
5×11
Spring 2015
CSE 373: Data Structures and Algorithms

23. CSE 373: Data Structures and Algorithms
Are we done?
The pattern seems pretty clear
Is there any reason to think it changes?
But we want something for any n ≥ 1
A mathematical approach is skeptical
Spring 2015
CSE 373: Data Structures and Algorithms

24. CSE 373: Data Structures and Algorithms
Are we done?
The pattern seems pretty clear
Is there any reason to think it changes?
But we want something for any n ≥ 1
A mathematical approach is skeptical
All we know is n(n+1)/2 works for 7 to 10
We must prove the formula works in all cases
A rigorous proof
Spring 2015
CSE 373: Data Structures and Algorithms

25. CSE 373: Data Structures and Algorithms
Proof by Induction
Prove the formula works for all cases.
Induction proofs have four components:
The thing you want to prove, e.g., sum of integers from 1 to n = n(n+1)/2
The base case (usually "let n = 1"),
The assumption step (“assume true for n = k")
The induction step (“now let n = k + 1").
n and k are just variables!
Spring 2015
CSE 373: Data Structures and Algorithms

26. CSE 373: Data Structures and Algorithms
Proof by induction
P(n) = sum of integers from 1 to n
We need to do
Base case
Assumption
Induction step
n and k are just variables!
prove for P(1)
assume for P(k)
show for P(k+1)
Spring 2015
CSE 373: Data Structures and Algorithms

27. CSE 373: Data Structures and Algorithms
Proof by induction
P(n) = sum of integers from 1 to n
We need to do
Base case
Assumption
Induction step
prove for P(1)
assume for P(k)
show for P(k+1) …
Spring 2015
CSE 373: Data Structures and Algorithms

28. CSE 373: Data Structures and Algorithms
Proof by induction
What we are trying to prove: P(n) = n(n+1)/2
Base case
P(1) = 1
1(1+1)/2 = 1(2)/2 = 1(1) = 1 ✓
Spring 2015
CSE 373: Data Structures and Algorithms

29. CSE 373: Data Structures and Algorithms
Proof by induction
What we are trying to prove: P(n) = n(n+1)/2
Assume true for k: P(k) = k(k+1)/2
Induction step:
Now consider P(k+1)
= … + k + (k+1)
Spring 2015
CSE 373: Data Structures and Algorithms

30. CSE 373: Data Structures and Algorithms
Proof by induction
What we are trying to prove: P(n) = n(n+1)/2
Assume true for k: P(k) = k(k+1)/2
Induction step:
Now consider P(k+1)
= … + k + (k+1)
= k(k+1)/2 + (k+1)
Spring 2015
CSE 373: Data Structures and Algorithms

31. CSE 373: Data Structures and Algorithms
Proof by induction
What we are trying to prove: P(n) = n(n+1)/2
Assume true for k: P(k) = k(k+1)/2
Induction step:
Now consider P(k+1)
= … + k + (k+1)
= k(k+1)/2 + (k+1)
= k(k+1)/2 + 2(k+1)/2
Spring 2015
CSE 373: Data Structures and Algorithms

32. CSE 373: Data Structures and Algorithms
Proof by induction
What we are trying to prove: P(n) = n(n+1)/2
Assume true for k: P(k) = k(k+1)/2
Induction step:
Now consider P(k+1)
= … + k + (k+1)
= k(k+1)/2 + (k+1)
= k(k+1)/2 + 2(k+1)/2
= (k(k+1) + 2(k+1))/2
Spring 2015
CSE 373: Data Structures and Algorithms

33. CSE 373: Data Structures and Algorithms
Proof by induction
What we are trying to prove: P(n) = n(n+1)/2
Assume true for k: P(k) = k(k+1)/2
Induction step:
Now consider P(k+1)
= … + k + (k+1)
= k(k+1)/2 + (k+1)
= k(k+1)/2 + 2(k+1)/2
= (k+1)(k+2)/2
= (k(k+1) + 2(k+1))/2
Spring 2015
CSE 373: Data Structures and Algorithms

34. CSE 373: Data Structures and Algorithms
Proof by induction
What we are trying to prove: P(n) = n(n+1)/2
Assume true for k: P(k) = k(k+1)/2
Induction step:
Now consider P(k+1)
= … + k + (k+1)
= k(k+1)/2 + (k+1)
= k(k+1)/2 + 2(k+1)/2
= (k+1)(k+2)/2
= (k(k+1) + 2(k+1))/2
Spring 2015
CSE 373: Data Structures and Algorithms

35. CSE 373: Data Structures and Algorithms
Proof by induction
What we are trying to prove: P(n) = n(n+1)/2
Assume true for k: P(k) = k(k+1)/2
Induction step:
Now consider P(k+1)
= … + k + (k+1)
= k(k+1)/2 + (k+1)
= k(k+1)/2 + 2(k+1)/2
= (k+1)(k+2)/2 ✓
= (k(k+1) + 2(k+1))/2
= (k+1)((k+1)+1)/2
Spring 2015
CSE 373: Data Structures and Algorithms

36. CSE 373: Data Structures and Algorithms
Proof by induction
What we are trying to prove: P(n) = n(n+1)/2
Assume true for k: P(k) = k(k+1)/2
Induction step:
Now consider P(k+1)
= … + k + (k+1)
= k(k+1)/2 + (k+1)
= k(k+1)/2 + 2(k+1)/2
= (k+1)(k+2)/2 ✓
= (k(k+1) + 2(k+1))/2
= (k+1)((k+1)+1)/2
Spring 2015
CSE 373: Data Structures and Algorithms

37. CSE 373: Data Structures and Algorithms
We’re done!
P(n) = sum of integers from 1 to n
We have shown
Base case
Assumption
Induction step
Success: we have proved that P(n) is true for any n ≥ 1 
proved for P(1)
assumed for P(k)
proved for P(k+1)
Spring 2015
CSE 373: Data Structures and Algorithms

38. CSE 373: Data Structures and Algorithms
Another one to try
What is the sum of the first n powers of 2?
n-1
k = 1: 20 = 1
k = 2: = = 3
k = 3: = = 7
k = 4: = = 15
For general n, the sum is 2n - 1
Spring 2015
CSE 373: Data Structures and Algorithms

39. How to prove it
P(n) = “the sum of the first n powers of 2 (starting at 0) is 2n-1”
Theorem: P(n) holds for all n ≥ 1
Proof: By induction on n
Base case: n=1. Sum of first 1 power of 2 is 20 , which equals 1 =
Inductive case:
Assume the sum of the first k powers of 2 is 2k-1
Show the sum of the first (k+1) powers of 2 is 2k+1-1
Spring 2015
CSE 373: Data Structures and Algorithms

40. CSE 373: Data Structures and Algorithms
How to prove it
The sum of the first k+1 powers of 2 is
(k-1) + 2k
= 2(2k) -1 = 2k+1 - 1
sum of the first k powers of 2
by inductive hypothesis
= 2k - 1
+ 2k
Spring 2015
CSE 373: Data Structures and Algorithms

41. CSE 373: Data Structures and Algorithms
Conclusion
Mathematical induction is a technique for proving something is true for all integers starting from a small one, usually 0 or 1.
A proof consists of three parts:
1. Prove it for the base case.
2. Assume it for some integer k.
3. With that assumption, show it holds for k+1
It can be used for complexity and correctness analyses.
Spring 2015
CSE 373: Data Structures and Algorithms

42. End of Inductive Proofs!
Spring 2015
CSE 373: Data Structures and Algorithms

43. Powers of 2
A bit is 0 or 1 (just two different “letters” or “symbols”)
A sequence of n bits can represent 2n distinct things
For example, the numbers 0 through 2n-1
210 is 1024 (“about a thousand”, kilo in CSE speak)
220 is “about a million”, mega in CSE speak
230 is “about a billion”, giga in CSE speak
Java: an int is 32 bits and signed, so “max int” is “about 2 billion”
a long is 64 bits and signed, so “max long” is 263-1
Spring 2015
CSE 373: Data Structures and Algorithms

44. Therefore… Could give a unique id to…
Every person in the U.S. with 29 bits
Every person in the world with 33 bits
Every person to have ever lived with 38 bits (estimate)
Every atom in the universe with bits
So if a password is 128 bits long and randomly generated,
do you think you could guess it?
Spring 2015
CSE 373: Data Structures and Algorithms